Symmetry and Asymmetry

• Many objects are built symmetrically
• Paired structures and serially homologous body parts are organized in a symmetric way

• Bilateral symmetry: structures arranged in either side of the body mid-line (axis of symmetry)

• Symmetry poses some challenges for morphometric data analysis

Types of Bilateral Symmetry

• Two main types of bilateral symmetry:
• Matching symmetry: pairs of structures are found across the mid-line of the object (e.g. insect wings, tetrapod limbs etc)
• Object symmetry: single structure with internal structural symmetry (e.g. structures in vertebrate skull)

The Problem with Symmetrical Objects

• Landmark positions are not independent in symmetric objects
• Lack of statistical independence for paired landmarks
• Some dimensions of shape space have little variance
• SSCP matrices become singular (or nearly so)
  • This causes issues with parametric statistical hypothesis-testing methods (‘divide’ by zero)

• The problem becomes worse as objects become more symmetric!

The Problem with Symmetrical Objects

• The data in reality have less dimensions, so use less data. How?
  • Digitize only half structure
  • Reflect structure and take the average: make a symmetrical structure

• Problem: using only \(\small\frac{1}{2}\) structure can introduce asymmetry, because the midline is not “anchored” as such

• Solution: use entire structure and RRPP for statistical evaluation!

• As we’ve seen this week, RRPP, combined with robust summary test statistics, does not require inverting (nearly) singular covariance matrices, so model evaluation is not compromised by this issue.

Symmetric Objects: Example

• Note the large number of shape dimensions with no variation. Much of this is due to bilateral symmetry!

The Biology of Asymmetry

• Different types of asymmetry are thought to have a biological meaning

• Directional asymmetry: consistent difference skewed towards one of the sides (at the population level); thought to reflect difference in use, e.g. fiddler crabs feeding vs. fighting displays

• Fluctuating asymmetry: thought to reflect instability during development (genetic, environmental etc)
• Long debate about the meaning of FA

Types of Symmetry

• Perfect symmetry is rare in biology; objects nearly always have some degree of asymmetry
• Three main types of asymmetry have been described, based on the statistical distribution of asymmetry in the population

• 1: Directional: consistent \(\small{(R-L)}\) differences with one side always larger than the other: \(\small{\overline{(R-L)}\neq0}\)
• 2: Fluctuating: small random \(\small{(R-L)}\) deviations with \(\small{\overline{(R-L)}=0}\)
• 3: Antisymmetry: consistent \(\small{(R-L)}\) differences, but larger side random

Analysis of Symmetry: General Procedure

• The presence of various types of asymmetry may be evaluated using one of several factorial ANOVA designs
• Traits need to be quantified multiple times, so that measurement error may be quantified
• This allows one to evaluate if the (normally small) differences between sides are actually “real” or due to measurement error (ME)

• ANOVA then often set up as:
• 

Extensions to GM Shape Data

• Procedure
  • 1: Digitize right and left structures (or R and L sides), possibly multiple times for ME
  • 2: Reflect one side to match the other (and relabel landmarks for object symmetry)
  • 3: GPA + projection to tangent space
  • 4: Factorial ANOVA – SS for Individual, SS Side (DA), SS Side x Individual (FA)
  • 5: Assess model factors via permutation approaches

Klingenberg & McIntyre (1998). Evolution.; Klingenberg et al. (2002). Evolution.

Decomposition of Asymmetry Component

• One can approach the problem more theoretically (rather than ‘procedurally’)
  • ANOVA effects respresent Sums of Squares, which correspond to various asymmetry components
  • Therefore, decomposing SST (total shape variation) obtained via \(\small{D}_Proc\) provides variance components for both DA and FA

\[\small{SST}=\sum^n_1{D}^2_{(X_i,Y_i)}=nD^2_{(\overline{X},\overline{Y})}+\sum^n_1{D}^2_{(X_i-\overline{X},Y_i-\overline{Y})}\]

• where: \(\small{nD}^2_{(\overline{X},\overline{Y})}\) represents the Directional Asymmetry component

• and: \(\small\sum^n_1{D}^2_{(X_i-\overline{X},Y_i-\overline{Y})}\) represents the Fluctuating Asymmetry component

Mardia et al. (2000). Biometrika.

Matching Symmetry: Example

Object Symmetry: Example

Other Types of Asymmetry

• Many other types of (a)symmetry are present in biological data
• How can we quantify asymmetry for these structures?

Symmetry Groups

• One can approach the problem by defining Symmetry Groups

• Symmetry groups: transformations that leave the data invariant

• e.g.: bilateral symmetry = reflection across the midline

• Symmetry groups define transformations such that there are invariances in those symmetric ‘dimensions’

• These groups describe the ways in which symmetry can be defined, and thus quantified for more complex structures (e.g., radial symmetry)

Savriama and Klingenberg. (2011). BMC Evol. Biol.

Symmetry: Extended Protocol

• For bilateral asymmetry: transformation = reflection
• For other types, decompose asymmetry to the components defining the symmetry type based on the appropriate symmetry groups

Symmetry: Extended Protocol

• For bilateral asymmetry: transformation = reflection
• For other types, decompose asymmetry to the components defining the symmetry type based on the appropriate symmetry groups

• Example: Radial Symmetry
  • Digitize all p components
  • GPA + projection
  • Decompose shape variation into symmetry components, quantify and evaluate

Savriama and Klingenberg. (2011). BMC Evol. Biol.

Complex Symmetry: Example

• Symmetry Analysis in corals

A Note on Geometric Transformations

• Any captured shape difference that follows a geometric rule can be modelled using the Procrustes paradigm
• We can account for undesired effects
• We can model shapes and other effects separately

• Articulations frequently cause this kind of effects
• Shape effects + positional effects

Geometric Transformations: Articulations

For articulated structures, several solutions exist - Fixing the angle in all specimens through a mathematical transformation - Separating the subsets to analyse separately, etc.

Adams. (1999). Evol. Ecol. Res.

Articulation Standardization: Example

• Standardize some data for relative jaw position

Another Note on Geometric Transformations

• Considering ‘fixing’ articulation angles in reverse results in a different pattern motion analysis
  • For shape devoid of movement, one fixes the angle.
  • For quantifying movement, one compares shapes across the motion path interested
• The latter forms a trajectory that describes the motion path
• It is represented by a sequence of shapes (a trajectory) in shape space

NOTE: see Phenotypic Trajectory Analysis lecture for analytical details

Symmetry: Summary

• Symmetry causes redundancy in shape data
• Can account for symmetry effects analytically
• Asymmetry: biological significance
• FA: developmental instability
• DA: adaptation?
• Use ANOVA procedures to separate variation within (asymmetry) and among individuals
• Extensions to different types of symmetry
• Geometric transformations in general: any effect that follows a geometric rule can be taken into account, partialed out, or targeted in the Procrustes framework